Cohomology of real analytic coherent sheaves

Let $$M$$ be a real analytic variety (if someone is concerned about distinction between "real analytic spaces" and "real analytic varieties" in real analytic geometry, let's assume that $$M$$ is both "variety" and "space"). I was sure it is well-known that higher cohomology of any real analytic coherent sheaf over $$M$$ vanish.

The argument is standard: by Grauert, a real analytic variety is the set of real points in a Stein variety $$M_{C}$$ with an anticomplex involution $$v$$. A coherent sheaf over $$M$$ is the set of $$v$$-invariant sections of a $$v$$-equivariant coherent sheaf $$F_{C}$$ on $$M_{C}$$, and $$F_C$$ has vanishing cohomology because $$M_C$$ is Stein.

Recently I needed a reference to this fact, and I could not find it. I would be extremely grateful for a reference!

• For smooth analytic manifolds, the reference is Proposition 2.3 in Atiyah and Hirzebruch's "Analytic cycles on complex manifolds", sciencedirect.com/science/article/pii/0040938362900940 – Grisha Papayanov Dec 7 at 23:35
• Many thanks. They don't seem to use the smoothness, in fact, the argument is almost literally the same as I gave. Still, the reference to the full strength statement would be extremely helpful. – Misha Verbitsky Dec 8 at 13:27
• That's because Grauert's paper On Levi's problem and the imbedding of real-analytic manifolds (sorry about russian version, the english one is behind paywall) which they are citing deals with smooth analytic manifolds. Do you know the reference for existence of complexification in a non-smooth case? – Grisha Papayanov Dec 8 at 13:43
• Sure: for non-smooth varieties it is actually a definition, see Guaraldo, F., Macri, P., Tancredi, A., {\em Topics on real analytic spaces}, Advanced lectures in mathematics, Braunschweig: F. Vieweg, 1986. – Misha Verbitsky Dec 8 at 14:12
• Theoreme 3 on page 89 is the statement about cohomology – Grisha Papayanov Dec 8 at 22:09

For a non-smooth case, I don't know the general reference, but Theoreme 3 in Henri Cartan's paper Variétés analytiques réelles et variétés analytiques complexes establishes theorems A and B for coherent (supports of coherent analytic sheaves) analytic subvarieties of $$\mathbb{R}^n$$